Coefficients \(a_{lm}, \tilde{a}_{lm}, c_l\)  and \(b_l\)  are determined so that the method is consistent of a given order. In particular, in addition to the order conditions specific to each sub-method, the coefficients should respect coupling conditions.
Here, we consider a variant of the IMEX method proposed in [1]}, whose coefficients are presented in the Butcher tableaux reported in Tables REF  and REF  for the explicit and implicit method, respectively, where \(\gamma =2 - \sqrt{2}.\) 
The coefficients of the explicit method were proposed in [1]}, while the implicit method, also employed in the same paper, coincides indeed for the above choice of \(\gamma \)  with the TR-BDF2 method proposed in [3]}, [4]} and applied to the Euler equations in [5]}. Notice that, even though we focus here on this specific second order method, the same strategy we outline is applicable to a generic DIRK method. In particular, higher order methods could be considered for coupling to high order spatial discretization, even though the effective overall accuracy would be limited by the splitting procedure if gravity and viscous terms are present.
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